Methods for correcting inhomogeneities of the magnetic field in a spatial volume provided for example in a cavity of a magnetic structure for a nuclear magnetic resonance imaging machine exist that are known and widely used.
Such method is based on the following theory disclosed with reference to a system of cartesian and three-dimensional coordinates.
The magnetic field in a spatial volume for each component has to satisfy Laplace's equation. Thus in three-dimensional space described by cartesian coordinates, components according to diretrixes of coordinate system have to satisfy following Laplace's equations:∇2Bx=0; ∇2By=0; ∇2Bz=0  (1)where Bx, By and Bz are components of the magnetic field along axes x, y and z of the coordinate system.
Thus it is possible to carry out a series expansion based on a set of orthogonal functions fn with n=1, . . . , ∞, which series for each x, y and z components is described by the following equation:
                    B        =                                            ∑                              i                =                0                            N                        ⁢                                          a                i                            ⁢                              f                i                                              =                                                    a                0                            ⁢                              f                0                                      +                                          ∑                                  i                  =                  1                                N                            ⁢                                                a                  i                                ⁢                                  f                  i                                                                                        (        2        )            
Apart from zeroth coefficients that are constants, components of the expansion having coefficients from 1 to N order describe inhomogeneities of the magnetic field along the direction of the corresponding coordinate.
Therefore for each component Bx, By and Bz of the magnetic field it is possible to define an inhomogeneity vector {right arrow over (a)} whose components are composed of coefficients a1, . . . aN of the series expansion and whose modulus corresponds to the measured inhomogeneity.
If a three-dimensional or two-dimensional grid of positioning points is defined outside the spatial volume wherein the magnetic field is examined and if a magnetic dipole is positioned in a positioning point of said grid, then even the magnetic field of the magnetic dipole has to satisfy Laplace's equation for each components and similarly to what has been made for the magnetic field under examination in the spatial volume it is possible to determine an effect vector of said dipole describing the dipole effect on the magnetic field in the spatial volume that is on one components of the magnetic field, components of said effect vector are composed of 1 to N order coefficients of a series expansion of one of components of the the magnetic field of said dipole based on the same set of orthogonal functions fn.
Therefore for each component of the magnetic field in the spatial volume it is possible to define a corresponding effect vector describing the effect of the corresponding component of the magnetic field of the dipole on the corresponding component of the magnetic field in the spatial volume.
In this case the magnetic field of dipole D in the positioning point p of the grid will be mathematically described for each components according to three directions of axes of the cartesian system defining the equation space
                              D          p                =                                            ∑                              i                =                0                            N                        ⁢                                          d                i                            ⁢                              f                i                                              =                                                    d                0                            ⁢                              f                0                                      +                                          ∑                                  i                  =                  1                                N                            ⁢                                                d                  i                                ⁢                                  f                  i                                                                                        (        3        )            
the effect vector {right arrow over (d)}p of dipole D in positioning point p will be therefore composed of components d1,p, . . . , dN,p.
Considering that m positioning points on three-dimensional or two-dimensional grid are all taken by a dipole, the effect of this dipole set on magnetic field inhomogeneities in spatial region is defined as the linear combination of effect vectors of all individual dipoles provided on the positioning grid and that is such effect will be provided by the the following linear combination:
                              ∑                      p            =            1                    M                ⁢                              c            p                    ⁢                                    d              →                        p                                              (        4        )            
where cp are linear combination coefficients. Since said dipoles have to generate a magnetic field whose effects are exactly contrary to magnetic field inhomogeneities in spatial volume, the ideal solution still for each component of the magnetic field along three axes of the coordinate system is given by the equation:
                                                                                  ∑                                  p                  =                  1                                M                            ⁢                                                c                  p                                ⁢                                                      d                    →                                    p                                                      +                          a              →                                                =        0                            (        5        )            
The solution of this equation leads to determine positions on three-dimensional or two-dimensional grid, the number of dipoles as well as to determine the magnetic moment of each dipoles necessary for compensating inhomogeneities of the magnetic field in said spactial volume obviously with respect to the corresponding component of the magnetic field along one of three axes. Positions are given by index P while coefficient cp is proportional to magnetic moment value that each dipole having a coefficient different from zero and position p in the grid of positioning points must have in order to achieve the compensation of inhomogeneities of the magnetic field in the volume where it is examined.
In practice said condition is never reached and therefore there is provided to determine the number of dipoles, charge values and specific positions on the grid that minimize the difference of equation (5).
By truncating the series expansions for a certain value of the index N, the number of components of vectors remains a finite one as well as by providing a three-dimensional or two-dimensional grid with a finite number M of positioning point p of correcting dipoles.
Although this method gave considerable results, it has some drawbacks. Particularly the known method does not minimize the number of dipoles and therefore positioning points thereof or the norm of magnetic moment modulus of the compensation dipole set.
There have been made attempts for reducing the computational effort and the number of dipoles and/or the total magnetic moment of the dipole set by introducing a symmetry-based approach that can help in defining repetetive models for positioning point subsets in delimited areas of the positioning grid. However the reduction of the number of dipoles and/or of the total magnetic moment of the set thereof and of the computational effort is not satisfactory.
This is due to the fact the use of the above mentioned method for determining dipoles can lead to combinations of dipoles that are close one with the other having reverse polarity and approximately the same magnetic moment modulus and whose effects partially suppress so that instead of having a solution providing a single dipole with a certain magnetic moment there are provided two dipoles or three dipoles with reverse polarities one with respect to the other and whose effects partially suppress causing a compensating effect that is very similar to the one of a single dipole in a different position.